Problem: A group of adults and kids went to see a movie. Tickets cost $$7.00$ each for adults and $$4.50$ each for kids, and the group paid $$50.00$ in total. There were $6$ fewer adults than kids in the group. Find the number of adults and kids in the group.
Solution: Let $x$ equal the number of adults and $y$ equal the number of kids. The system of equations is then: ${7x+4.5y = 50}$ ${x = y-6}$ Solve for $x$ and $y$ using substitution. Since $x$ has already been solved for, substitute ${y-6}$ for $x$ in the first equation. ${7}{(y-6)}{+ 4.5y = 50}$ Simplify and solve for $y$ $ 7y-42 + 4.5y = 50 $ $ 11.5y-42 = 50 $ $ 11.5y = 92 $ $ y = \dfrac{92}{11.5} $ ${y = 8}$ Now that you know ${y = 8}$ , plug it back into ${x = y-6}$ to find $x$ ${x = }{(8)}{ - 6}$ ${x = 2}$ You can also plug ${y = 8}$ into ${7x+4.5y = 50}$ and get the same answer for $x$ ${7x + 4.5}{(8)}{= 50}$ ${x = 2}$ There were $2$ adults and $8$ kids.